Fast Acceleration and Velocity Estimation forWideband Stretching LFM Radars Based
on Mutual Bias CorrectionYixiong Zhang , Xiufang Chen, Huawei Xu , Xiao-Ping Zhang , Senior Member, IEEE, and Feng Qi
Abstract—Linear frequency modulated (LFM) signal iswidely used in wideband radars, where stretch processingis commonly adopted to reduce the processing bandwidth.In this paper, we propose a fast acceleration and velocityestimation method for wideband stretching LFM radars basedon mutual bias correction. In this method, cross-correlationis carried out on adjacent stretched echoes to integratescattering energy. By deriving the relationship between thetwo-dimensional (2D) spectrum of the cross-correlationresultand the motion parameters, it is found that the frequency ofthe 2D spectrum peak is determined by the acceleration andthe velocity. We first estimate a coarse acceleration using thefrequency of the 2D spectrum peak on the slow time, and thenconstruct a compensation function to estimate the coarse velocity. To obtain high precision estimation, we develop amutual bias correction (MBC) process to reduce the bias of the coarse estimates. Experimental results show that theproposed method achieves much smaller root mean square error (RMSE) with comparable computational cost, comparedto existing state-of-the-art methods.
Index Terms— LFM, motion parameter estimation.
I. INTRODUCTION
THE wideband linear frequency modulation (LFM) signalis widely used in modern radar systems [1]–[3]. Due to
the higher range resolution, the motions of the target oftencause range migration (RM) and Doppler frequency migration(DFM) effects in the received signals, resulting in qualitydecrease of inverse synthetic aperture radar (ISAR) imaging.
Manuscript received January 15, 2020; revised March 12, 2020;accepted March 15, 2020. Date of publication April 3, 2020; date ofcurrent version July 6, 2020. This work was supported in part by thePrincipal Foundation of Xiamen University under Grant 20720180075,in part by the Natural Sciences and Engineering Research Council ofCanada (NSERC) under Grant RGPIN239031, in part by the ChongqingResearch Program of Basic Research and Frontier Technology underGrant cstc2016jcyjA0301, in part by the NSFC Project under Grant61405001, and in part by the Venture & Innovation Support Program forChongqing Overseas Returnees under Grant cx2018121. The asso-ciate editor coordinating the review of this article and approving itfor publication was Prof. Piotr J. Samczynski. (Corresponding author:Yixiong Zhang.)
Yixiong Zhang and Xiufang Chen are with the School of Informatics,Xiamen University, Xiamen 361005, China (e-mail: [email protected];[email protected]).
Huawei Xu is with the Technology Center Department, NetEase, Inc.,Hangzhou 310052, China (e-mail: [email protected]).
Xiao-Ping Zhang is with the Department of Electrical and ComputerEngineering, Ryerson University, Toronto, ON M5B 2K3, Canada (e-mail:[email protected]).
Feng Qi is with the Shenyang Institute of Automation, Chinese Acad-emy of Sciences, Shenyang 110016, China (e-mail: [email protected]).
Digital Object Identifier 10.1109/JSEN.2020.2983839
To improve the performances of ISAR imaging, target trackingand identification, high precision motion estimation is impor-tant in wideband LFM radars [4]–[13]. In these widebandLFM radar systems, stretch processing is commonly utilizedto reduce the sampling rate of analog-to-digital (ADC) and theprocessing bandwidth.
In the literature, most of the traditional motion estimationmethods in ISAR imaging belong to the searching methods.Among them, envelope correlation method and entropy mini-mization method are most commonly used. One of the mainadvantages of envelope correlation method is the small compu-tational cost [14], [15]. By calculating the envelope correlationcoefficient of the one-dimensional (1D) range profiles of theadjacent echoes and the moving distance between neighboringpulse envelope, the velocity can be estimated. The entropyof the image reflects the average amount of information ofthe image. The simpler the image, the smaller the entropy.In [16]–[20], focusing degree of range profile is measuredby the image entropy and the velocity can be estimatedby searching the target speed. However, both of these twomethods ignore the phase information. Therefore, they wouldhave low estimation precision under low signal-to-noise-ratio(SNR) case.
Abatzoglou et al. use a maximum likelihood estimationmethod to estimate target distance, velocity and acceler-ation [19], and achieve the optimal theoretical precision.
Multi-pulses measurement method based on general RadonFourier transform (GRFT) is proposed in [21]–[24] on thebasis of [19]. In GRFT method, multi-dimensional search isused to search the motion parameters of the target. The veloc-ity can be obtained from the maximum energy position whichis accumulated by GRFT. For this method, the Cramer-Raolower bound (CRLB) of velocity can be reached, but the multi-dimensional search leads to huge computational complexity.
To reduce the computational complexity, a non-searchingestimation method based on adjacent cross-correlation func-tion (ACCF) is proposed to estimate the motion parame-ters of targets [25], [26]. Compared to the GRFT method,the ACCF method can obtain similar estimation precision withmuch lower computational cost. The ACCF method performscross-correlation on adjacent pulses which are obtained aftermatch filtering. In other words, it is suitable for uncom-pressed received signal instead of stretched signal. Therefore,the ACCF method cannot be directly applied to the stretchingsystem. To solve the fast motion estimation problem for wide-band stretching systems, a fast non-searching motion parame-ters estimation method based on cross-correlation of adjacentechoes (CCAE) is proposed in [27]. The CCAE method canbe applied to both the uncompressed signal and the stretchedsignal. However, the CCAE method obtains acceleration byperforming differential operation on the estimated velocities,and the accuracy is relatively low.
In this paper, to estimate the acceleration and velocity of thetarget for wideband stretching systems, a fast non-searchingestimation method based on mutual bias correction (MBC)between acceleration and velocity is proposed. In the proposedmethod, to integrate the energy from different scatterers of thetarget, cross-correlation is applied to the adjacent pulse echoes.After deriving the relationship between the 2D spectrum ofthe cross-correlation results and the motion parameters, it isinvestigated that the acceleration and the velocity are coupledin the frequency of the 2D spectrum peak along the slow time.To estimate the motion parameters, we firstly use the peakfrequency of the 2D spectrum to obtain a coarse acceleration.Then a coarse velocity is estimated by compensating thecross-correlation results with the coarse acceleration. Finally,a mutual bias correction process between the accelerationand the velocity is developed to obtain accurate estimation.Compared to the ACCF method, the proposed MBC estimationmethod can achieve better root mean square error (RMSE)performances, with much lower computational cost.
Although the ACCF and CCAE methods also use cross-correlation to estimate motion parameters, the proposedMBC method is substantially different from them, which isdescribed in detail as follows. First, the cross-correlationsof the ACCF method and the proposed method are appliedto different signal types. In the ACCF method, matchedfiltering is first applied to two adjacent uncompressed echoesby the FFT operation, and then, a conjugate multiplication isperformed on the matched filtered results. Finally, to estimatethe motion parameters, the correlation result is converted tothe time domain by an IFFT operation. In the proposed MBCmethod, the cross-correlation operation is directly carried outon the uncompressed or stretched signals, and then 2D FFT
operation is performed on the correlation results. As a result,the computational cost of MBC method is lower than ACCFmethod. More importantly, the MBC method can be directlyincorporated into the stretching system.
When compared to the CCAE method [27], the proposedMBC method in this paper is improved extensively. In [27],the acceleration is obtained by applying differential operationon the estimated frequency of the 1D FFT result. This isan incoherent integration method, and it only uses threepulse echoes. However, for the MBC method of this paper,the acceleration is estimated after 2D FFT operation andbias correction. It is a coherent integration method and canuse multiple pulse echoes to integrate energy. Therefore, theestimation precision of MBC method is much higher than thatof CCAE method [27].
The rest of this paper is organized as follows. In Section II,the model of the wideband stretched LFM signal is presented.Section III introduces the proposed method in detail. InSection IV, the performance of the proposed method is ana-lyzed. Experiments are conducted and discussed in Section V.Finally, we conclude this paper in Section VI.
II. SIGNAL MODEL
The proposed method can be applied to uncompressedsignal or stretched signal. In this paper, we mainly discussthe stretched signal model. The transmitted wideband LFMsignal can be written as:
str (t) = rect (t/T ) exp(
jπγ t2)
exp ( j2π fct) , (1)
where T is the pulse width of the transmitted signal, fc is the
center frequency, t is the fast time and rect (u) ={
1, |u| ≤ 12
0, |u| > 12
is the rectangle pulse function. γ = BT denotes the chirp rate
of LFM signal, where B is the swept bandwidth of the LFMsignal.
The point-scattering model is used in this paper. The uncom-pressed wideband echo signal can be written as:
s′re(t, tm) =
P−1∑p=0
A prect
[t − τp (tm)
T
]
× exp(
jπγ(t − τp (tm)
)2+ j2π fc(t − τp (tm)
))+ ω0(t, tm)
= sre(t, tm) + ω0(t, tm), (2)
where sre(t, tm) is the signal part reflected from the target.P denotes the number of scatterers, A p is the scatteringcoefficient of the p-th scatterer and τp denotes the round triptime delay from radar to the p-th scatterer. tm = mTr is theslow time, where m is the pulse number, and Tr is the pulserepetition interval (PRI). ω0(t, tm) is the white Gaussian noisewith mean zero and variance σ 2. Assuming the amplitudes ofall the scatterers are the same, i.e. A p = A, the SNR of theuncompressed signal is A2/σ 2.
The local reference signal for stretch processing can bewritten as:
slo(t) = rect[t/T
]exp
(− jπγ t2
)exp (− j2π fct) , (3)
ZHANG et al.: FAST ACCELERATION AND VELOCITY ESTIMATION FOR WIDEBAND STRETCHING LFM RADARS 8685
Fig. 1. Stretch processing.
where T is the length of the receiving window or the referencesignal, and it is usually larger than T to ensure that it can coverthe received signal. The difference value Tw = T − T is thelength of range window.
Due to the large bandwidth, the analog-to-digital (ADC)with very high sampling rate is needed to sample the uncom-pressed wideband LFM signal, and the size of the sampleddata is very large. Therefore, stretch processing is commonlyadopted to reduce the processing bandwidth in wideband LFMradars. The stretch processing is shown in Fig. 1. The leftpart describes the time-frequency image of the local refer-ence signal and the uncompressed echo signal. After stretchprocessing, the wideband signal is converted to narrowbandsignal, of which the range profiles are shown in the right partof Fig. 1. It is easy to see that the spectral peaks in the rangedomain correspond to locations of the scatterers.
The stretched signal can be obtained by multiplication:
s′st (t, tm) = s′
re (t, tm) · slo (t)
=P−1∑p=0
A prect
[t − τp (tm)
T
]exp
(− j2πγ τp (tm) t)
× exp(− j2π fcτp (tm)
) × exp(
jπγ τ 2p (tm)
)+ ω1(t, tm)
= sst (t, tm) + ω1(t, tm), (4)
where ω1(t, tm) = ω0(t, tm) · sre(t, tm) is the noise afterstretching and sst (t, tm) is the stretched target signal. As canbe seen from Fig. 1, the maximum bandwidth of the stretchedsignal depends on the size of range window. And the sizeof range window is generally much smaller than the size ofreceiving window. Hence the bandwidth of the stretched echosignal is much smaller than that of local reference signal. Thus,low sampling rate can be applied to reduce the computationalcost after stretch processing.
The time delay of the target scatterer is defined as:
τp (tm) = 2rp (tm) /c = 2
c(rp + v ptm + 1
2apt2
m), (5)
where rp is the distance from the p-th scatterer to the centerof receiving window. v p and ap correspond to the velocityand acceleration, respectively. Different scatterers of the targetcan be considered to have the same velocity and acceleration,
i.e., v p = vq = v and ap = aq = a, where v and a denotethe velocity and acceleration of the target, respectively.
Given the parameters of radar signal ( fc, B , γ , Tr , T ),the objective of the proposed method is to estimate the velocityv and the acceleration a of the target for wideband stretchingLFM signal.
III. THE PROPOSED MBC ESTIMATION METHOD
In radar systems, within the interval of two transmittedpulses, the motion parameter change of the target is generallysmall. Therefore, the range profiles of adjacent echoes arehighly correlated. As we know, the stretch processing canbe used to estimate the time delay of the two input LFMsignals. Inspired by this, we carry out the stretching operation(i.e., cross-correlation) to the adjacent echoes, and obtain thedistance difference of the target between adjacent pulses. Thenwe perform 2D-FFT operation on the cross-correlation resultsand derive the relationship between the 2D spectrum and themotion parameters.
A. Cross-Correlation of Adjacent EchoesIn this subsection, we derive the relationship between
the frequency of the 2D spectrum peak and the motionparameters after performing cross-correlation and 2D FFToperation on the stretched pulse echoes. The cross-correlationmethod comes from the idea of “stretch processing”, in whichthe frequency of the stretched signal is determined by thetime delay difference between adjacent pulse echoes andcan be used to estimate the motion parameters. The cross-correlation operation (i.e. conjugate multiplication) betweenthe stretched echo at tm+1 and the stretched echo at tm can berepresented as
sac (t, tm) = sst (t, tm) · s∗st (t, tm+1)
= sse (t, tm) + scr (t, tm) + ω2(t, tm), (6)
where ∗ represents the conjugate operation, sse(t, tm) is theautocorrelation term from the same scatterers, scr (t, tm) isthe cross-correlation term induced by different scatterers,and ω2(t, tm) is the noise after cross-correlation. They arerepresented as follows.
There are two rectangular windows in (7) and (8) withthe same size. But their positions are inconsistent, due to thedifferent time delay. After multiplication, the non-overlappingpart of the two rectangular windows results in a loss of signalenergy. The time length of the non-overlapping part can beexpressed as
�tp,q = τp(tm+1) − τq(tm) = 2
c((rp − rq ) + vTr )), (10)
where rp −rq denotes the distance between any two scatterers.It is seen that �tp,q is determined by the target displacementbetween two pulse echoes. Suppose the pulse duration isT = 1ms, the pulse interval is Tr = 10ms, the length ofthe target is 50m and the instantaneous velocity of the targetis 5000m/s. Then we can calculate that the maximum value of�tp,q is 200
c . In this case, the ratio between the non-overlappingpart and the pulse duration length is 200
cT = 0.0667%. Thus,the influence of this signal loss can be negligible.
After performing Fourier transformation on the cross-correlation result along the fast time, we have
W2( f, tm) is the FFT result of ω2(t, tm). As shown in (12)and (13), both Sse( f, tm) and Scr ( f, tm) compose of multiplesinc functions. Different sinc functions have different peakpositions. It can be seen that the peak position of the sinc
function is f pq = 2γ (rp−rq)c + E p
m . As the target is in the far
Fig. 2. The target scatterers distribute normally along the RLOS.
.
Fig. 3. The 1D spectrum of the cross-correlation result (SNR = −20dB).
field, different scatterers of the target can be considered tohave the same velocity and acceleration, i.e., v p = vq = v,ap = aq = a, where v and a denote the velocity andthe acceleration of the target, respectively. When p = q ,the result is the autocorrelation term, shown in (12). In thiscase, the energy of different scatterers is focused on the samespectrum peak. On the other hand, when p �= q , differentscatterers have different distance values, i.e., rp �= rq . Theenergy disperses at different spectrum positions. As a result,there are multiple sub-peaks and a main peak in Sac ( f, tm).In general, the amplitude of the main peak is much biggerthan that of the sub-peaks. The influence of the cross-terms isdiscussed as follows.
In the 1D spectrum of the cross-correlation result, the energyof the cross-terms (13) will accumulate for the same value ofrp − rq . When the target scatterers distributes normally alongthe radar line of sight (RLOS), the amplitude of the cross-termwith rp+1−rp or rp−rp+1 (p and p+1 are adjacent scatterers)will be the highest. This case is illustrated in Fig. 2, wherethe distance between two adjacent scatterers is δ. In this case,the amplitude of the cross-term with
∣∣rp − rq∣∣ = δ would be
the highest among the cross-terms. Suppose the echo strengthof all the scatterers are the same, i.e. A p = Aq = A. Then theamplitude of the self-term is P A2. And the amplitude of thehighest cross-term is (P − 1) A2, which is smaller than theamplitude of the self-term.
Following, we give an example to show the influence ofthe cross terms. The radar parameters are set as: the carrierfrequency fc = 9GHz, the pulse interval Tr = 10ms, thebandwidth B = 1GHz, the pulse duration T = 100μs, andthe chirp rate γ = B/T = 1013. The target parameters are
ZHANG et al.: FAST ACCELERATION AND VELOCITY ESTIMATION FOR WIDEBAND STRETCHING LFM RADARS 8687
as follows. The number of scatterers P = 4, the distancebetween two adjacent scatterers δ = 0.6m, and the velocityand acceleration are v = 100m/s and a = 10m/s2 respec-tively. Fig. 3 shows the 1D spectrum of the cross-correlationresult, of which the SNR of the uncompressed echo signal is−20dB. It can be seen that the amplitude of the highest crossterm is smaller than the self-term. For the ease of analysis,the following derivation is mainly concentrated on Sse ( f, tm).
B. The 2D Spectrum of the Cross-Correlation ResultUsing v p = v and ap = a, we re-write E p
m of (14) as,
E pm = Em = 2γ
(vTr + 1
2 aT 2r + aTr tm
)c
. (15)
Em reflects the time delay difference of the target betweenpulse tm and tm+1.
By expanding (12), we have
Sse( f, tm) =P−1∑p=0
A2psinc
[T
(f − E p
m)]
× ex p(
j2π(
H0 + H1tm − H2t2m − H3t3
m
)),
(16)
H0 = fc
c
(2vTr + aT 2
r
),
− 2γ
c2
(2rpvTr+v2T 2
r +rpaT 2r +vaT 3
r + 1
4a2T 2
r
),
(17)
H1 = 2 fcaTr
c+ 2γ
c2
(2v2Tr +2rpaTr +3vaT 2
r +a2T 3r
),
(18)
H2 = 2γ
c2
(3vaTr + 3
2a2T 2
r
), (19)
H3 = 2γ
c2 a2Tr . (20)
H0 is the constant phase term. H1, H2, and H3 denotes thefirst, second, and third order terms of tm . The instantaneousfrequency of Sse( f, tm) on slow time can be represented as
ftm = H1 − 2H2tm − 3H3t2m . (21)
Furthermore, H1 can be re-written as
H1 = H1_0 + H1_1 + H1_2 + H1_3 + H1_4. (22)
H1_0 = 2 fcaTr
c, (23)
H1_1 = 4γ v2Tr
c2 , (24)
H1_2 = 4γ rpaTr
c2 , (25)
H1_3 = 6γ vaT 2r
c2 , (26)
H1_4 = 2γ a2T 3r
c2 . (27)
It is observed that H2tm = mT 2r
(H1_3 + H1_4
)and H3t2
m =m2 H1_4. Tr is the PRI. In general, Tr � 1 s, e.g. 10ms. m isthe pulse number, which is smaller than 10 in our method.
Therefore, H2tm � H1_3 + H1_4. Also, due to the item c2
in the denominator, H1_1, H1_2, H1_3 and H1_4 are generallysmaller than H1_0. And we only need to compare H1_2, H1_3and H1_4 with H1_1.
In radar tracking system, the center of the receiving windowis normally close to the center of the target, and the scatterer ofthe target is also close to the center of the receiving window,i.e., rp is very small. Besides, the target velocity is usuallymuch larger than the target acceleration, and Tr � 1 s. Thus,H1_1 is much larger than H1_2, H1_3 and H1_4. From theabove analysis, among the components of ftm , H2tm , H3t2
m ,H1_2, H1_3 and H1_4 can be neglected, and the instantaneousfrequency can be approximated as
ftm ≈ H1_0 + H1_1 = 2 fcaTr
c+ 4γ v2Tr
c2 . (28)
Then we have,
Sse ( f, tm) ≈ sinc [T ( f − Em)]
× exp
(j2π
(2 fc
ca − 4γ
c2 v2)
Tr tm
)
×P−1∑p=0
A2p exp ( j2πφ) , (29)
φ = fc
c(2vTr + aT 2
r )
− 2γ
c2
(2rpvTr +v2T 2
r +rpaT 2r +vaT 3
r + 1
4a2T 4
r
),
(30)
where exp ( j2πφ) is the constant phase term. From (29), it canbe seen that the phase term is a linear function of tm . Whenthe change of position of the sinc function in Sse( f, tm) doesnot exceed half of an frequency resolution, multiple cross-correlation results can be used to integrate energy and improvethe estimation accuracy.
Suppose we use N pulse echoes, then there are N −1 cross-correlation results, i.e., Em, Em+1, · · · , Em+N−2. Similar to(14), the time delay difference between tm+N−2 and tm+N−1,i.e., Em+N−2 can be written as,
Em+N−2 = γ(τp (tm+N−1) − τp (tm+N−2)
)= 2γ
(vTr + 1
2 a(2N − 3)T 2r + aTr tm
)c
. (31)
Investigating (15) and (31), their first and third itemsare the same. The second item changes and may leadto position change of the sinc function. If the positionchange is smaller than half of a frequency resolution cell,migration across cells will not take place. In other words,the non-migration condition of acceleration estimation can berepresented as,
(Em+N−2 − Em) <γ
2B⇒ |a| <
c
4(N − 2)BT 2r
. (32)
If B is 1 GHz, Tr is 10 ms and a is 80 m/s2 which is closeto the maximum acceleration of the rocket, then N should notbe more than 11.
Fig. 4. The spectra of the cross-correlation results (9 pulses). (a) 2D spectrum for SNR = 2dB, v = 100m/s, a = 0m/s2. (b) 2D spectrum forSNR = −10dB, v = 100m/s, a = 10m/s2. (c) The slice of (a) on slow time. (d) The slice of (b) on slow time.
Performing FFT on (29) along the slow time, we have,
Sse(
f, ftm
) ≈ sinc [T ( f − Em)]
× sinc
[2Tr
(ftm − 2 fcTr
ca + 4γ Tr
c2 v2)]
×P−1∑p=0
A2p exp ( j2πφ) . (33)
From (33) we can obtain the frequency of peak position onthe slow time.
ftm =(
2 fc
ca − 4γ
c2 v2)
Tr . (34)
Eq. (34) represents the relationship between the spectrumpeak of the cross-correlation result with the motion parameters.In (34), fc, γ , and Tr are known parameters, and ftm is theestimated peak frequency. The velocity v and acceleration aare coupled in the spectrum peak ftm . In the next subsections,we will find a solution to estimate them from (34).
The 2D spectrum of the cross-correlation results is illus-trated in Fig. 4. Nine pulses are used to integrate energy.In order to demonstrate the influence of the cross-term andnoise on the sidelobes, two sets of parameters are used toobtain the 2D spectra. In Fig. 4(a), the SNR of the uncom-pressed echo signal is 2dB, and the velocity and accelerationof the target are 100 m/s and 0 m/s2 respectively. In Fig. 4(b),SNR is −10dB, and the velocity and acceleration of the targetare 100 m/s and 10 m/s2 respectively. From Fig. 4(a) andFig. 4(b), it is seen that the energy of different pulse echoesis focused on the spectrum peak along the slow time.
It is noticed that there are two types of sidelobes in the 2Dspectrum. The sidelobes on the fast time are mainly causedby the cross-terms of the correlation results and the windowfunction (such as the rectangle shape function we used), whichcan be seen from Fig. 3. On the other hand, the sidelobes onthe slow time are caused by the window function (rectangleshape function) and the second and higher order phase termof tm in the self-term of the cross-correlation result ((16)).However, the influence of the higher order phase term is
ZHANG et al.: FAST ACCELERATION AND VELOCITY ESTIMATION FOR WIDEBAND STRETCHING LFM RADARS 8689
tiny. In the self-term (16), when the acceleration of the targetis zero, the second and third order terms (H2 (19) and H3(20)) with regard to tm also equal to zero. Fig. 4(c) andFig. 4(d) respectively shows the spectrum slice of slow timeextracted from Fig. 4(a) and Fig. 4(b). It is seen that when theacceleration is zero, there are still sidelobes on the slow time.Also, as the second and higher order term of tm have someinfluence on the amplitudes of the sidelobes, the amplitudes ofthe sidelobes with acceleration (Fig. 4(d)) are a bit higher thanthat without acceleration (Fig. 4(c)). In addition, by comparingFig. 4(a) and Fig. 4(b), we can see that the amplitudes of thesidelobes are similar for different SNRs (2dB and −10dB).This demonstrates that the sidelobes are not caused by noise.In a word, the sidelobes are caused by the window function,the cross terms and the high order phase term of tm .
C. Coarse Estimation of Acceleration and VelocityIn (34), although the acceleration and the velocity are
coupled in the spectrum peak, the first term is much greaterthan the second term. So ftm can be used to coarsely estimatethe acceleration, i.e., the coarse acceleration. That is
a = c ftm
2 fcTr= a − 2γ
c fcv2. (35)
There is an error term in the coarse acceleration. As shownin (35), the coarse acceleration is affected by the veloc-ity, and the bias of coarse acceleration can be defined as�a = − 2γ
c fcv2.
As shown in (29), if the acceleration of the target is known,the 1D spectrum peak position of the cross-correlation resultis only determined by the velocity. Therefore, to eliminate theaTr tm related terms in (29), we use the coarse acceleration toconstruct a compensation function as follows.
scomp (t, tm) = exp
(− j
4πγ
caTr tmt
)×exp
(− j
4π fc
caTr tm
)
= exp
(− j
4πγ
caTr tmt + j
8πγ 2
c2 fcv2Tr tmt
)
× exp
(− j
4π fc
caTr tm + j
8πγ
c2 v2Tr tm
).
(36)
According to (29), the self-term of the cross-correlationresult is re-written as
sse(t, tm) ≈ exp
(j4πγ t
c
(vTr + 1
2aT 2
r + aTr tm
))
× exp
(j2π
(2 fc
ca − 4γ
c2 v2)
Tr tm
)
×P−1∑p=0
A2p exp ( j2πφ) . (37)
Multiplying scomp (t, tm) by sse (t, tm), we have,
sseco (t, tm) ≈ exp
(j4πγ
c
(vTr + 1
2aT 2
r + 2γ
c fcv2Tr tm
)t
)
×P−1∑p=0
A2p exp ( j2πφ) . (38)
After converting the fast time domain into frequencydomain, we have,
Sseco ( f, tm)
≈ sinc
[T
(f − 2γ
c
(vTr + 1
2aT 2
r + 2γ
c fcv2Tr tm
))]
×P−1∑p=0
A2p exp ( j2πφ) . (39)
To accumulate signal energy, we can also use N pulseechoes, and there will be N − 1 cross-correlation results inthe slow time domain for Sseco ( f, tm): tm = 0, tm+1 = Tr ,. . ., tm+N−2 = (N − 2) Tr . Adding them together, then wehave,
Scompsum ( f, tm)
≈ sinc
[T
(f − 2γ
c
(vTr
1
2aT 2
r
))] P−1∑p=0
A2p
+ sinc
[T
(f − 2γ
c
(vTr + 1
2aT 2
r − �aT 2r
))] P−1∑p=0
A2p
+ . . . + sinc
[T
(f − 2γ
c
(vTr + 1
2aT 2
r
− (N − 2) �aT 2r
))] P−1∑p=0
A2p exp ( j2πφ). (40)
As mentioned above, if 2γ (N−2)�aT 2r
c < γ2B , that is,
2(N−2)γc fc
v2T 2r < c
4B , migration across cells can be avoided.Therefore, the maximum accumulated pulse number can beobtained:
2 (N − 2) γ
c fcv2T 2
r = c
4B⇒ N = c2 fc
8γ Bv2T 2r
+ 2. (41)
For example, the radar parameters are set as following:B = 1 GHz, Tr = 10 ms, fc = 9 GHz, T = 100 μs andv = 1000 m/s. Then, N ≈ 101. In general, the magnitudeorder of c
4B is between 10−2 and 10−1. In common widebandLFM radars, B/ fc ≈ 10−1 and cT > 104, so 2γ
c fc< 10−5 � 1.
If PRI, the accumulated pulse numbers N and v satisfy certainconditions, such as the magnitude order of both vTr and Nis 10, 2(N−2)γ
c fcv2T 2
r < 10−2 < c4B can be satisfied and
migration across cells will not occur between Sseco ( f, tm) andSseco ( f, tm+N−2).
There are two conditions for N . In (32), the constrainton N is to ensure that there is no range migration betweenthe cross-correlation results before acceleration compensation.And the condition of (41) is to ensure that there is norange migration between the cross-correlation results afteracceleration compensation. As we know, the range migrationhas been relieved by compensating the acceleration. Thus,the constraint of (32) is stricter than that of (41). Therefore,both of the two conditions for N should be satisfied, and thefinal constraint for N is the condition (32).
As a comparison, the addition of the N − 1 cross-correlation results without acceleration compensation is
Fig. 5. The 1D spectrums of the addition of multiple cross-correlationresults (N = 9) before and after acceleration compensation.
written as,
Sorisum ( f, tm) = Sse ( f, tm) + Sse ( f, tm+1)
+ · · · + Sse ( f, tm+N−2) . (42)
For single cross-correlation result, the spectrum after accel-eration compensation is similar to that before accelerationcompensation (Fig. 3), because the coarse acceleration is usedto compensate the phase term of the cross-correlation result.For the addition of multiple cross-correlation results (N = 9,i.e., 8 cross-correlation results), the 1D spectra before andafter acceleration compensation are shown in Fig. 5. It can beseen that the amplitude of the main peak after accelerationcompensation (see (40)) is much higher than that beforeacceleration compensation (see (42)). This implies that theenergy can be well accumulated and the SNR can be improvedafter acceleration compensation.
In (40), there are N−1 cross-correlation results Sseco( f, tm),where tm = 0, Tr , 2Tr · · · (N−2)Tr . The peak frequency of thefirst sinc function Sseco( f, tm = 0) is ftm = 2γ
c
(vTr + 1
2 aT 2r
).
The peak frequency of the last sinc function Sseco( f, tm =(N − 2)Tr ) is ftm+N−2 = 2γ
c
(vTr + 1
2 aT 2r − (N − 2)�aT 2
r
).
Then, according to the symmetry property of sinc function,the peak position of (40) appears at the center of the peakpositions of the N−1 sinc functions, i.e., the average of ftm andftm+N−2 .
ft = 1
2
(ftm + ftm+N−2
)= 2γ
c
(vTr + 1
2aT 2
r − (N − 2)
2�aT 2
r
)
= 2γ
c
(vTr + 1
2aT 2
r + (N − 2) γ
c fcv2T 2
r
). (43)
In (43), the third term is much smaller than the first term. Inaddition, the estimated coarse acceleration in (35) can be usedto eliminate a. Thus we can estimate the coarse velocity as
v = c ft
2γ Tr− aTr
2= v + (N − 1) γ Tr
c fcv2. (44)
The bias of the coarse velocity can be written as− N−1
2 Tr�a. If the magnitude order of vTr and N is 102 and10, respectively, the velocity bias is less than one percent, thatis N−1
2 Tr · 2γc fc
v2 < 0.01v.
D. Correction of the BiasAs the correlation coefficient between the absolute values
of the acceleration bias and the velocity bias is positive,the coarse velocity can be used to offset the bias of accel-eration, and vice versa. Add 2γ
c fcv2 to (35) and the estimate of
acceleration is corrected as:
ac = a − 2γ
c fcv2 + 2γ
c fcv2
= a + 2γ Tr
c fc(− (N − 1) �a · v
+ �a2Tr (N − 1)2
4). (45)
ac is the corrected acceleration after correctionand the bias of the corrected acceleration is �ac =2γ Trc fc
(− (N − 1) �a · v + �a2Tr (N−1)2
4 ). In order to ensurethat |�ac| < |�a|, the magnitude order of vTr N should beless than 105. In general, the magnitude order of N is lessthan 102 and the magnitude order of vTr is less than 103.Therefore, the above condition can be satisfied in normalcases.
With the corrected acceleration, repeat the process from (36)to (44), we can get:
vc = c ft
2γ Tr− acTr
2= v − N − 1
2�acTr . (46)
vc is the corrected velocity after correction. If the condi-tion |�ac| < |�a| can be satisfied, then
∣∣− N−12 Tr�ac
∣∣ <∣∣− N−12 Tr�a
∣∣. It means that the bias of final velocity is alsodecreased. After the correction process, the estimated precisionof acceleration and velocity are both improved. We name thiscorrection process as mutual bias correction (MBC), wherethe biases of acceleration and velocity are corrected by eachother. It should be noticed that, repeating the process from(45) to (46) for multiple times does not reduce the bias further.On the contrary, the bias increase instead in next iteration inour experiment. Because the bias form of ac is different fromthat of a, it is meaningless to construct another item similarto 2γ
c fcv2
c . Therefore, it is difficult to construct a specific itemto decrease the bias of acceleration estimates in next iteration.
From the above analysis, it is seen that the improvementfor the acceleration bias after correction is |�ac − �a|, andthe improvement for the velocity bias after correction is|�vc − �v| = ∣∣− N−1
2 Tr∣∣ |�ac − �a|. Tr is the pulse rep-
etition interval, N is the number of accumulated pulses.In general,
∣∣− N−12 Tr
∣∣ is less than 1. Thus, |�vc − �v | <|�ac − �a|. This implies that the improvement for the veloc-ity estimation is smaller than that for acceleration estimation.
Fig. 6 shows the steps of the proposed method in detail.There are three parts in this flowchart. In the first part,the cross-correlation operation and the 2D-FFT are carriedout on adjacent pulse echoes. In the second part, the coarse
ZHANG et al.: FAST ACCELERATION AND VELOCITY ESTIMATION FOR WIDEBAND STRETCHING LFM RADARS 8691
Fig. 6. The flowchart of the proposed method.
acceleration and velocity are estimated. The mutual biascorrection between the acceleration and velocity is performedin the third part.
IV. PERFORMANCE ANALYSIS
In this section, we analyze the theoretical RMSE of thecoarse acceleration. Having in mind from (2) and (4), Theuncompressed and stretched echo signals are respectivelyrepresented as:
s′re (t, tm) = sre (t, tm) + ω0 (t, tm) , (47)
s′st (t, tm) = sst (t, tm) + ω1 (t, tm) . (48)
where sre and sst denote the target signal part of the uncom-pressed and stretched echo, respectively. The two noises areboth additive white Gaussian noise. Assume that the noisevariance in (47) is σ 2 and the noise bandwidth is equal tothe signal bandwidth B . Because the bandwidth of low passfilter is generally equal to sampling frequency fs , the noisebandwidth of the stretched echo signal is also equal to thesampling frequency. The SNR of the uncompressed echosignal is defined as SN Rre = A2/σ 2 [28], [29], where
TABLE IRADAR PARAMETERS
A denotes the amplitude of the uncompressed signal. Then,the SNR of the stretched echo signal can be written as:
SN Rst = A2
fsB σ 2
= B
fsSN Rre . (49)
Due to the decrease of noise bandwidth, the SNR ofstretched echo signal is B
fstimes higher than that of original
echo signal. The noise after cross-correlation is:
ω2 (t, tm) = sst (t, tm) · ω∗1 (t, tm+1)
+ s∗st (t, tm+1) · ω1 (t, tm)
+ ω1 (t, tm) · ω∗1 (t, tm+1) . (50)
In (50), sst (t, tm) ω∗1 (t, tm+1) and s∗
st (t, tm+1) ω1 (t, tm) aresubject to Gaussian distribution. ω1 (t, tm) ω∗
1 (t, tm+1) followsdouble Gaussian distribution. It is difficult to analyze thisdistribution exactly. But it can be approximated to be Gaussiandistribution [30]. As a result, the mean values of these threenoise items in (50) are all zero, and variance are A2σ 2 fs
B ,A2σ 2 fs
B and σ 4 f 2s
B2 , respectively. Thus, we can obtain the SNRof sac (t, tm) ((6)):
SN Rac,t = A4
fsB 2A2σ 2 + f 2
sB2 σ 4
= SN Rst
2 + 1S N Rst
. (51)
After applying FFT operation to sac (t, tm), the SNR ofsac ( f, tm) can be written as:
SN Rac, f = N f · SN Rac,t = N f · SN Rst
2 + 1S N Rst
, (52)
where N f is the number of samples at fast time.According to (35), the coarse acceleration is estimated by
using the frequency of the 2D spectrum along the slow time,
that is a = c ftm2 fcTr
. Thus, the RMSE of the estimates of sinusoidfrequency can be used to evaluate the performance of coarseacceleration [31]. The RMSE of the frequency estimation fora complex sinusoid signal with unknown frequency, amplitudeand phase can be expressed as:
RM SE ftm =√
6
4π2SN R∇2 M(M2 − 1
) , (53)
where ∇ is the sampling interval and M is the number ofsampling points.
Along the slow time, the sampling interval is the pulserepetition interval, i.e.,∇ = Tr . The number of samplingpoints equals to the number of cross-correlation results, that isM = Ns . Based on the relationship between the acceleration
Fig. 7. The performances of the proposed MBC method with different parameters. (a) The velocity RMSEs. (b) The acceleration RMSEs.
Fig. 8. The bias before and after correction. (a) The velocity bias. (b) The acceleration bias.
and the estimated frequency, the theoretical lower bound ofthe RMSE of the coarse acceleration is represented as:
RM SEa = c
2 fcTr
√6
4π2SN Rac, f T 2r Ns
(N2
s − 1)
= c
2 fcT 2r
√6
4π2SN Rac,t N f Ns(N2
s − 1) . (54)
where, Ns = N −1 because N pulses are used for accelerationestimation.
In the proposed method, the velocity estimation is obtainedby averaging the frequencies of N − 1 items, i.e., sac (t, tm),. . ., sac (t, tm+N−2). They are all double Gaussian func-tions. After adding them together, the distribution function isunknown. Therefore, it is intractable to derive the close-formRMSEs for the coarse velocity, the corrected acceleration andthe corrected velocity. Numerical experiments are carried outto verify the performances of these parameters.
V. SIMULATION
In this section, the parameters of the target are set as:the distance r0 = 10000 m, the velocity v0 = 100 m/s and
Fig. 9. The theoretical bias with different accumulated pulse numbers.
the acceleration a0 = 10 m/s2. The radar parameters areshown in TABLE I. We utilize the Newton iteration method toestimate the peak spectrum position [32]. 1000 Monte Carlosimulations are performed for each SNR case.
ZHANG et al.: FAST ACCELERATION AND VELOCITY ESTIMATION FOR WIDEBAND STRETCHING LFM RADARS 8693
Fig. 10. The RMSE performances of MBC method with different accumulated pulse numbers. (a) The RMSE of estimated velocity. (b) The RMSEof estimated acceleration.
Fig. 11. The estimated mean values under different accumulated pulse numbers. (a) The velocity mean values. (b) The acceleration mean values.
A. The Performances of the Proposed Method WithDifferent Bandwidth and PRI
First, we evaluate the RMSE of the proposed MBC methodwith different bandwidths and PRIs, under different SNRconditions. Fig. 7(a) and Fig. 7(b) show the RMSE resultsof velocity and acceleration respectively. It is observed thatthe RMSE of velocity decreases when the bandwidth or thePRI increases. The RMSE performance of the case withB = 1 GHz and Tr = 10 ms has the best performance. Whilefor the RMSE of acceleration, the performance is independentof the bandwidth, which is in accordance with (54). TheRMSE of acceleration decreases with the increase of PRI.
As we know, the number of FFT points along the slowtime is equal to the number of cross correlation results(i.e., N − 1). The precision of frequency estimation dependson the number of FFT points. To obtain high estimation pre-cision, we use nine pulses to test the estimation performancesbefore and after correction. The comparisons of bias beforeand after correction are shown in Fig. 8(a) and Fig. 8(b),
respectively. It can be seen that the biases of estimates arewell reduced after correction, especially for the acceleration(shown in Fig. 8(b)).
B. The Performances of the Proposed Method WithDifferent Accumulated Pulse Numbers
The theoretical biases of acceleration and velocityare �ac = 2γ Tr
c fc(− (N − 1) �a · v + �a2Tr (N−1)2
4 ) and
�vc = − N−12 �acTr , respectively. With the pre-setting motion
parameters, the absolute values of theoretical bias of the cor-rected velocity and acceleration are shown in Fig. 9. It can beseen that the bias of acceleration is approximately linear withthe number of accumulated pulses. The orders of the velocityand acceleration bias are 10−5. Therefore, the accumulatedpulse number has little effect on the biases.
The RMSE performances of the proposed MBC methodwith different accumulated pulse numbers are evaluated.The velocity and acceleration RMSE results are shown inFig. 10(a) and Fig. 10(b) respectively. With the increase of
Fig. 12. The estimated RMSEs of MBC, ACCF and CCAE. (a) The velocity RMSEs. (b) The acceleration RMSEs.
TABLE IITIME COSTS OF THE PROPOSED METHOD WITH
DIFFERENT PULSE NUMBER
pulse number, the RMSE performance of the proposed MBCmethod becomes better. Fig. 11(a) and Fig. 11(b) show themean values of the estimated velocity and acceleration fordifferent accumulated pulse numbers. In general, the biaseswill decrease with the increase of SNR, but the biases jitterwith SNR in Fig. 11(a) and Fig. 11(b). The reason is thatthe number of the second FFT is equal to the number ofaccumulated pulses. The more the number of accumulatedpulses is, the higher the accuracy of the frequency estimationwill be. Therefore, the more the accumulated pulse number is,the less serious the jitter will be (e.g. the case of nine pulsesin Fig. 11). With the increase of pulse number, although thetheoretical bias will increase (as shown in Fig. 9), the actualbias becomes smaller, due to more accumulated energy.Besides, the pulse number should also satisfy the conditionof (32).
In this simulation, the total time costs of the 1000 MonteCarlo simulations for all of the SNR cases are listed inTable II. It can be seen that the computational cost is not verysensitive to the accumulated pulse number.
C. Comparison With CCAE and ACCFIn this subsection, the performance of the proposed MBC
estimation method is compared with the CCAE and ACCFmethod. In this experiment, the uncompressed and stretchedsignals are used to test MBC and CCAE methods. Theuncompressed (UR) signals are for the ACCF method, whichcan only be applied to the UR signals. For these threemethods, three pulse echoes are used in this experiment.It should be noted that the two methods, i.e., 2D-FFT and
differential operation, are used to estimate the accelerationin ACCF. For ACCF_FFT, the acceleration is estimated afterperforming 2D-FFT on the cross-correlation results. Whilefor ACCF_Diff, the acceleration is estimated by performingdifferential operation on the estimated velocities, similar toCCAE. As a comparison, the acceleration RMSE of CCAE iswritten as [27]:
RM SEa,CCAE =√
2c
2γ T 2r
√√√√ 6
4π2SN Rac,t T 2f N f
(N2
f − 1)
≈√
2c
2BT 2r
√6
4π2SN Rac,t N f, (55)
where T f is the sampling interval at fast time. By com-paring (54) and (55), it is clear that the accelerationRMSE of the proposed method is much smaller than thatof CCAE.
The performances of velocity and acceleration are comparedin Fig. 12(a) and Fig. 12(b), respectively. For CCAE and MBCmethods, the legend “stretched” or “uncompressed” meansthe type of data used. Because the energy is accumulated inthe ACCF_FFT method, the performance of the ACCF_FFTmethod is better than that of ACCF_Diff.
When using the uncompressed data, the estimation per-formances of ACCF_FFT and MBC are almost the same.However, the computational cost of ACCF_FFT is higher thanMBC, as it needs to perform more FFT/IFFT operations. Also,the estimation performance of ACCF_Diff is similar to thatof CCAE on uncompressed data. The most important is thatMBC and CCAE methods can be applied to stretched signal,which is commonly adpoted in modern wideband radars.By comparing MBC and CCAE, it is seen that MBC methodcan effectively improve the estimation precision. Besides, theperformances of MBC method are also better than that ofACCF_FFT, as the SNR of the stretched signal is higher thanthat of the UR signal.
For the cross-correlation based estimation method, whetherit is MBC/CCAE or ACCF, when the amplitude of theself-term of the correlation result is smaller than the noise
ZHANG et al.: FAST ACCELERATION AND VELOCITY ESTIMATION FOR WIDEBAND STRETCHING LFM RADARS 8695
TABLE IIITIME COSTS OF CCAE, THE PROPOSED METHOD AND ACCF
TABLE IVFPGA RESOURCE USAGE OF THE PROPOSED METHOD AND CCAE
amplitude after cross-correlation, we cannot find the peak ofself-term correctly. Thus the estimated frequency would bewrong. This is known as the threshold SNR phenomenon. Forinstance, the threshold SNR is about −21dB in accelerationestimation by ACCF_FFT. When the SNR is smaller than−21dB, the RMSE does not deteriorate much, due to thatthe peak search along fast time is totally wrong. For highSNR cases, the noise variance is small and has little effect onthe performance. Therefore, the RMSE performances of theproposed MBC method and the ACCF_FFT are similar.
The total time costs of the 1000 Monte Carlo simulationsfor all of the SNR cases are listed in Table III. It can beseen that the time cost of CCAE is the least, as it use1D-FFT and simple differential operation. The time cost ofthe proposed MBC method is much smaller than that ofACCF. The data size of UR signal and the number of FFToperation are two main reasons for the high time cost of ACCF.More importantly, the proposed MBC method obtains the bestestimation precision among these methods.
To evaluate the real-time performance, we implemented theproposed MBC and the CCAE methods using Verilog HDLand verified them in FPGA, using Xilinx Kintex-7 XC7K480T.The FPGA resource usage MBC and CCAE are listed in IV.When the clock frequency is set as 200MHz, the averagetime costs of the MBC and CCAE method based on threepulses are about 50.23μs and 72.29μs, respectively. Therefore,the proposed method can estimate the target parameters in realtime when the PRI is larger than 75μs, which can be satisfiedin most of the radar systems.
VI. CONCLUSIONS
Motion parameters estimation is important to ISARimaging, target tracking and recognition in wideband LFMradars. In this paper, we proposed a fast acceleration andvelocity estimation method based on mutual bias correctionfor wideband stretching LFM radars. In the proposed method,cross-correlation is applied to the stretched echoes, and the1D spectrum of the cross-correlation result shows that thescatterering energy of the target is accumulated on the main
spectrum peak. By deriving the relationship between the2D spectrum of the cross-correlation results and the motionparameters, it is found that the acceleration and the velocityare coupled in the peak frequency of the 2D spectrum. We firstestimate a coarse acceleration using the peak frequency ofthe 2D spectrum, and then a coarse velocity is estimatedby compensating the coarse acceleration with the cross-correlation results. With acceleration compensation, the energyof multiple cross-correlation results can be integrated toimprove the accuracy of the coarse velocity. As the coarseacceleration and velocity are biased, we develop a mutual biascorrection between acceleration and velocity to obtain accurateestimation. The proposed MBC method can be applied tothe uncompressed or stretched signal. The theoretical RMSEof acceleration is derived in this paper. Compared toCCAE method, MBC method provides much better RMSEperformance with a bit higher computational cost. Whencompared to ACCF method, MBC method also providesbetter RMSE performance with much lower computationalcost.
In this paper, we do not consider the higher order motionparameters, such as jerk. When considering higher ordermotion parameters, it is intractable to obtain the analyticalexpression of the motion parameters from the spectrum infor-mation of the stretched signals. Thus it becomes difficult toestimate the motion parameters by using the proposed method.Nevertheless, the main contribution of the proposed method isthat it can estimate high precision velocity and accelerationinstantaneously using several pulse echoes. Besides, in manyscenes, the target motion can be approximated by velocity andacceleration in a short time. To estimate higher order motionparameters using wideband stretching LFM radars is an openproblem for our future work.
[2] F. Berizzi, E. D. Mese, M. Diani, and M. Martorella, “High-resolution ISAR imaging of maneuvering targets by means of therange instantaneous Doppler technique: Modeling and performanceanalysis,” IEEE Trans. Image Process., vol. 10, no. 12, pp. 1880–1890,Dec. 2001.
[3] S. Liu, T. Shan, Y. D. Zhang, R. Tao, and Y. Feng, “A fast algorithmfor multi-component LFM signal analysis exploiting segmented DPTand SDFrFT,” in Proc. IEEE Radar Conf. (RadarCon), May 2015,pp. 1139–1143.
[4] S. A. S. Werness, W. G. Carrara, L. S. Joyce, and D. B. Franczak,“Moving target imaging algorithm for SAR data,” IEEE Trans. Aerosp.Electron. Syst., vol. 26, no. 1, pp. 57–67, Jan. 1990.
[5] V. C. Chen and S. Qian, “Joint time-frequency transform for radar range-Doppler imaging,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2,pp. 486–499, Apr. 1998.
[6] C.-C. Chen and H. Andrews, “Target-motion-induced radar imaging,”IEEE Trans. Aerosp. Electron. Syst., vols. AES–16, no. 1, pp. 2–14,Jan. 1980.
[7] J. Zheng, H. Liu, G. Liao, T. Su, Z. Liu, and Q. H. Liu, “ISAR imagingof targets with complex motions based on a noise-resistant parameterestimation algorithm without nonuniform axis,” IEEE Sensors J., vol. 16,no. 8, pp. 2509–2518, Apr. 2016.
[8] A. S. Khwaja and X.-P. Zhang, “Sensitivity analysis of compressedsensing ISAR imaging to rotational acceleration rate mismatch,” in Proc.IEEE Int. Conf. Image Process., Sep. 2013, pp. 2349–2353.
[9] A. S. Khwaja and X.-P. Zhang, “Motion parameter estimationand focusing from SAR images based on sparse reconstruction,”IEEE Geosci. Remote Sens. Lett., vol. 11, no. 8, pp. 1350–1354,Aug. 2014.
[10] A. S. Khwaja and X.-P. Zhang, “Compressed sensing basedimage formation of SAR/ISAR data in presence of basis mis-match,” in Proc. 19th IEEE Int. Conf. Image Process., Sep. 2012,pp. 901–904.
[11] A. Khwaja and X.-P. Zhang, “Compressed sensing SAR moving targetimaging in the presence of basis mismatch,” in Proc. IEEE Int. Symp.Circuits Syst. (ISCAS), May 2013, pp. 1809–1812.
[12] A. S. Khwaja and X.-P. Zhang, “Compressed sensing ISAR reconstruc-tion in the presence of rotational acceleration,” IEEE J. Sel. TopicsAppl. Earth Observ. Remote Sens., vol. 7, no. 7, pp. 2957–2970,Apr. 2014.
[13] S. Liu et al., “Sparse discrete fractional Fourier transform and its appli-cations,” IEEE Trans. Signal Process., vol. 62, no. 24, pp. 6582–6595,Oct. 2014.
[14] J. Gao, X. Xu, and F. Su, “Range cell migration of SAR moving targetbased on envelope correlation method,” in Proc. 1st Int. Conf. Electron.Instrum. Inf. Syst. (EIIS), Jun. 2017, pp. 1–5.
[15] J. M. Munoz-Ferreras and F. Perez-Martinez, “Extended envelope corre-lation for range bin alignment in ISAR,” in Proc. IET Int. Conf. RadarSyst., 2007, pp. 1–5.
[16] L. Xi, L. Guosui, and J. Ni, “Autofocusing of ISAR images based onentropy minimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 35,no. 4, pp. 1240–1252, Oct. 1999.
[17] H.-R. Jeong, H.-T. Kim, and K.-T. Kim, “Application of subarrayaveraging and entropy minimization algorithm to stepped-frequencyISAR autofocus,” IEEE Trans. Antennas Propag., vol. 56, no. 4,pp. 1144–1154, Apr. 2008.
[18] J. Wang and X. Liu, “SAR minimum-entropy autofocus using anadaptive-order polynomial model,” IEEE Geosci. Remote Sens. Lett.,vol. 3, no. 4, pp. 512–516, Oct. 2006.
[19] T. J. Abatzoglou and G. O. Gheen, “Range, radial velocity, and acceler-ation MLE using radar LFM pulse train,” IEEE Trans. Aerosp. Electron.Syst., vol. 34, no. 4, pp. 1070–1083, Oct. 1998.
[20] R. P. Bocker, T. B. Henderson, S. A. Jones, and B. R. Frieden, “Newinverse synthetic aperture radar algorithm for translational motion com-pensation,” Proc. SPIE Int. Soc. Opt. Photon., vol. 1569, pp. 298–311,Oct. 1991.
[21] J. Xu, X.-G. Xia, S.-B. Peng, J. Yu, Y.-N. Peng, and L.-C. Qian,“Radar maneuvering target motion estimation based on generalizedradon-Fourier transform,” IEEE Trans. Signal Process., vol. 60, no. 12,pp. 6190–6201, Dec. 2012.
[22] J. Xu, J. Yu, Y.-N. Peng, and X.-G. Xia, “Radon-Fourier trans-form for radar target detection, I: Generalized Doppler filter bank,”IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 2, pp. 1186–1202,Apr. 2011.
[23] J. Xu, J. Yu, Y.-N. Peng, and X.-G. Xia, “Radon-Fourier transformfor radar target detection (II): Blind speed sidelobe suppression,”IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 4, pp. 2473–2489,Oct. 2011.
[24] J. Yu, J. Xu, Y.-N. Peng, and X.-G. Xia, “Radon-Fourier transformfor radar target detection (III): Optimality and fast implementations,”IEEE Trans. Aerosp. Electron. Syst., vol. 48, no. 2, pp. 991–1004,Apr. 2012.
[25] X. Li, G. Cui, W. Yi, and L. Kong, “A fast maneuveringtarget motion parameters estimation algorithm based on ACCF,”IEEE Signal Process. Lett., vol. 22, no. 3, pp. 270–274,Mar. 2015.
[26] X. Li, G. Cui, L. Kong, and W. Yi, “Fast non-searching methodfor maneuvering target detection and motion parameters estima-tion,” IEEE Trans. Signal Process., vol. 64, no. 9, pp. 2232–2244,May 2016.
[27] Y.-X. Zhang, R.-J. Hong, C.-F. Yang, Y.-J. Zhang, Z.-M. Deng, andS. Jin, “A fast motion parameters estimation method based on cross-correlation of adjacent echoes for wideband LFM radars,” Appl. Sci.,vol. 7, no. 5, p. 500, 2017.
[28] X.-G. Xia, “A quantitative analysis of SNR in the short-time Fouriertransform domain for multicomponent signals,” IEEE Trans. SignalProcess., vol. 46, no. 1, pp. 200–203, Jan. 1998.
[29] X.-G. Xia and V. C. Chen, “A quantitative SNR analysis for the pseudowigner-ville distribution,” IEEE Trans. Signal Process., vol. 47, no. 10,pp. 2891–2894, Oct. 1999.
[30] N. O’Donoughue and J. M. F. Moura, “On the product of independentcomplex gaussians,” IEEE Trans. Signal Process., vol. 60, no. 3,pp. 1050–1063, Mar. 2012.
[31] D. Rife and R. Boorstyn, “Single tone parameter estimation fromdiscrete-time observations,” IEEE Trans. Inf. Theory, vol. IT-20, no. 5,pp. 591–598, Sep. 1974.
[32] T. Abatzoglou, “A fast maximum likelihood algorithm for fre-quency estimation of a sinusoid based on Newton’s method,” IEEETrans. Acoust., Speech, Signal Process., vol. 33, no. 1, pp. 77–89,Feb. 1985.
Yixiong Zhang was born in Fujian, China,in 1981. He received the B.S. degree ininformation engineering and the Ph.D. degreein information and communication engineeringfrom Zhejiang University, Hangzhou, China,in 2003 and 2009, respectively. In 2009, he joinedthe School of Information Science and Engineer-ing, Xiamen University, Xiamen, China, where heis currently an Associate Professor. His currentresearch interests include signal detection andparameter estimation, inverse synthetic aperture
radar imaging, and hardware/software codesign of embedded systems.
Xiufang Chen was born in China, in 1993.She is currently pursuing the M.E. degree incommunication engineering with Xiamen Univer-sity, Xiamen, China. She has been involved inundergraduate research trainings with XiamenUniversity since 2018. Her interests include radarsignal processing and software development.
Huawei Xu was born in China, in 1993. Hereceived the B.S. and M.S. degrees in com-munication engineering from Xiamen University,in 2016 and 2019, respectively. He is a SoftwareEngineer with the Technology Center Depart-ment, NetEase, Inc. His interests include radar,game development, and software development.
Authorized licensed use limited to: Ryerson University Library. Downloaded on July 03,2020 at 16:58:13 UTC from IEEE Xplore. Restrictions apply.
ZHANG et al.: FAST ACCELERATION AND VELOCITY ESTIMATION FOR WIDEBAND STRETCHING LFM RADARS 8697
Xiao-Ping Zhang (Senior Member, IEEE)received the B.S. and Ph.D. degrees in elec-tronic engineering from Tsinghua Universityin 1992 and 1996, respectively, and the M.B.A.(Hons.) degree in finance, economics and entre-preneurship from the University of Chicago BoothSchool of Business, Chicago, IL, USA. SinceFall 2000, he has been with the Department ofElectrical and Computer Engineering, RyersonUniversity, Toronto, ON, Canada, where he is cur-rently a Professor and the Director of the Commu-
nication and Signal Processing Applications Laboratory. He has servedas the Program Director of Graduate Studies. He is cross-appointed tothe Finance Department, Ted Rogers School of Management, RyersonUniversity. He was a Visiting Scientist with the Research Laboratoryof Electronics, Massachusetts Institute of Technology, Cambridge, MA,USA, in 2015 and 2017. He is a frequent consultant for biotech com-panies and investment firms. He is the Co-Founder and the CEO forEidoSearch, an Ontario-based company offering a content-based searchand analysis engine for financial big data. His research interests includeimage and multimedia content analysis, statistical signal processing, sen-sor networks and electronic systems, machine learning, and applicationsin big data, finance, and marketing. He is a Senior Area Editor of the IEEETRANSACTIONS ON SIGNAL PROCESSING. He is/was an Associate Editor ofthe IEEE TRANSACTIONS ON IMAGE PROCESSING, the IEEE TRANSACTIONSON MULTIMEDIA, the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FORVIDEO TECHNOLOGY, the IEEE TRANSACTIONS ON SIGNAL PROCESSING,and the IEEE SIGNAL PROCESSING LETTERS.
Feng Qi received the bachelor’s degree fromZhejiang University, and the master’s and Ph.D.degrees from Katholieke Universiteit Leuven,Belgium, in 2005 and 2011, respectively. Since2005, he has been working on antennas andmillimeter wave imaging. Then, he joined RIKEN,Japan, by working on nonlinear optics. Since2014, he has also been doing research on THzradar in Goethe University, Germany, and theUniversity of Birmingham, U.K. In 2015, he joinedChinese Academy of Sciences as a Professor by
heading the Terahertz Imaging Lab in the Shenyang Institute of Automa-tion. His research interests include microwave, laser, and radar. Since2012, he has also been serving the Global Symposium on MillimeterWaves as a TPC member. He received the Best Paper Award in 2014.
Authorized licensed use limited to: Ryerson University Library. Downloaded on July 03,2020 at 16:58:13 UTC from IEEE Xplore. Restrictions apply.
